Lettau’s Definition of the Obukhov Stability Parameter
While Lettau (1939) adhered strictly to the nomenclature given by Prandtl and Richardson, the accessibility of the work under consideration here (Lettau 1949) is relatively difficult, since Lettau (1949) used primes for differentiation with respect to height and used the French accent grâve for turbulent fluctuations (while the present work follows the notation that is common today). Furthermore, Lettau’s definition for the covariance for non-isotropic turbulence (grey-shaded equations and the sign of fluxes and parameters are identical to the original reference)
is different from the other literature, where \(u^{*} \) is referred to as a friction velocity, and \(w^{*} \) as a mixing velocity, where \(u^{\prime }\) and \(w^{\prime}\) are turbulent fluctuations of the streamwise and vertical velocity components. According to the definition of isotropic turbulence (Taylor 1935), “In isotropic turbulence the average value of any function of the velocity components, defined in relation to a given set of axes, is unaltered if the axes of reference are rotated in any manner” and the associated assumption of an adiabatic atmosphere (subscript a) follows
and thus for the stress
Confusing in this context is the definition of the mixing length l with the local averaged vertical velocity component \( w^{*}\) in Lewis (1997) about the balloon experiments of Lettau and Schwerdtfeger, for the time interval τ (not stress!)
$$ l = w^{*} \tau = \mathop \int \limits_{t}^{t + \tau } w^{\prime } {\text{d}}t. $$
(6)
Lettau and Schwerdtfeger (1933a, b) used this approach to calculate the exchange coefficients by Hesselberg (1929) and Ertel (1932). However, they used the symbol \(\zeta^{*}\) and not \(w^{*}\). The symbol and the definition differ. In Lettau (1949), there is no reference to it.
For non-isotropic conditions, Lettau assumed that the ratio of mixing velocity and mixing length does not depend on the stratification
The difference between adiabatic conditions and conditions with existing stratification is described by the buoyancy force. The term
describes the vertical component of turbulent acceleration under adiabatic conditions. Thus follows
with Tm as absolute mean temperature in the surface layer. From Eqs. 7 and 8, it follows that
Lettau used the notation
and defined relationships between adiabatic and non-adiabatic conditions, for instance for the mixing length
He provided analogous relationships for the mixing velocity, the exchange coefficient (A, see Eq. 1), and the sensible heat flux (for which he used the symbol L instead of H in the following)
where cp is the specific heat at constant pressure. Besides that, he related x to the Richardson number as
At this point, Lettau introduced another dimensionless function of height, the adiabatic mixing velocity, and the sensible heat flux without further explanation (with the roughness length z0, see Fig. 1)
which is nearly identical to the Obukhov stability parameter z/L defined by Eq. 19 below. For adiabatic and isotropic conditions, \(w_{a}^{*}\) according to Eq. 5—see also Eq. 26a in Lettau (1949)—is equivalent to the friction velocity in today’s definition. From Eqs. 11 and 12 follows the relation between x and y,
Rossby and Montgomery (1935) used a similar approach to derive an equation for the vertical wind-speed profile, which means that they separated the adiabatic part from the stratification-influenced part. However, they did not separate the covariance \(\overline{{u^{\prime } w^{\prime } }}\) according to Eq. 4. Finally, they found the relationship
$$ \frac{{{\text{d}}u}}{{{\text{d}}z}} = \frac{1}{{\kappa \left( {z + z_{0} } \right)}}\sqrt {\frac{\tau }{\rho }} \sqrt {\frac{1}{2} + \frac{1}{2}\sqrt {1 + \frac{{4x^{2 } \left( {z + z_{0} } \right)^{2} }}{{\left( {\frac{1}{\kappa }\sqrt {\frac{\tau }{\rho }} } \right)^{2} }}} } . $$
(16)
With the definition by Rossby and Montgomery (1935)
$$ x^{2} = \beta \frac{g}{T} \frac{{{\text{d}}\theta }}{{{\text{d}}z}} = \beta \frac{g}{T} \frac{H}{{c_{p} \rho \kappa u_{*} \left( {z + z_{0} } \right)}} , $$
(17)
where β is an unknown dimensionless coefficient, it follows that
$$ \frac{{4x^{2 } \left( {z + z_{0} } \right)^{2} }}{{\left( {\frac{1}{\kappa }\sqrt {\frac{\tau }{\rho }} } \right)^{2} }} = 4 \beta \kappa \left( {z + z_{0} } \right) \frac{g}{T } \frac{{\frac{H}{{c_{p } \rho }}}}{{u_{*}^{3} }} \approx 4 \beta \kappa z \frac{g}{T } \frac{{\frac{H}{{c_{p } \rho }}}}{{u_{*}^{3} }} = 4 \beta \frac{z}{L} . $$
(18)
Rossby and Montgomery (1935) used Eq. 16 to calculate the deviation of the non-adiabatic wind profile from the logarithmic wind profile for nearly isothermal conditions, and Sverdrup (1936) did the same for non-isothermal conditions. But they did not separate the term on the right-hand side of Eq. 18 and investigate its properties.
According to the current nomenclature (including the negative sign of L for unstable conditions with H > 0, no distinction between the friction velocity and the mixing velocity, and adiabatic conditions), from Eq. 14 with z » z0 and from Eq. 18 with β = 0.25, what follows is exactly the Obukhov stability parameter with the Obukhov length L
$$ \frac{z}{L} = - \frac{{z \kappa \frac{g}{{T_{m} }} \frac{H}{{\rho c_{p} }}}}{{u_{*}^{3} }}. $$
(19)
However, this simplification is not in complete agreement with the mixing velocity according to Lettau (1949), because Eq. 5 assumes adiabatic conditions and isotropy, while in Eq. 19H ≠ 0 is possible. Businger (1955) already pointed out this problem in the discussion of Eq. 8. However, Obukhov (1946) and Monin and Obukhov (1954) give a range for z << L where the exchange conditions differ little from exchange conditions in a neutrally-stratified atmosphere independent of the magnitude of the sensible heat flux (sublayer of dynamic turbulence). Skeib (1980) (see also Foken and Skeib 1983) determined a critical height to which the assumption applies and which is 0.5 m to about 5 m depending on the stratification. Thus, it must be stated that the parameter y according to Lettau (1949) and z/L according to Obukhov (1946) are not exactly identical with respect to the definition, but are largely identical near the surface in the dynamical sublayer.
Monin and Obukhov (1954) also defined the Obukhov length as a function of the friction velocity, the sensible heat flux, and the buoyancy parameter \(g/T_{m} , \) but derived this relationship with the Π-theorem and not empirically. However, the relation between x and y or Ri and z/L was first obtained by Lettau (1949), earlier than Monin and Obukhov (1954), and is still used today with little numerical difference for stable stratification (Businger et al. 1971)
$$ {\raise0.7ex\hbox{$z$} \!\mathord{\left/ {\vphantom {z L}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$L$}}\, = Ri\quad{\text{for}}\,Ri < 0, $$
(20)
$$ {\raise0.7ex\hbox{$z$} \!\mathord{\left/ {\vphantom {z L}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$L$}} = \frac{Ri}{{1 - 5 Ri}} \quad{\text{for}}\,0 \le Ri \le 0.2 = Ri_{c} . $$
(21)
Obukhov’s Definition of the Obukhov Length
Obukhov (1946) also used Schmidt's exchange approach. Obukhov solved the problem of stratification by using a function of the Richardson number,
where in the neutral case
. It is interesting that Obukhov also began his derivation with two generalized velocities, the friction velocity (symbols according to Obukhov 1946)
and the "thermal" mixing velocity
with the sensible heat flux q. [Remark: "thermal" mixing velocity, in Russian: cкopocть пepeнoca тeплa; in the English translation (Obukhov 1971) a formulation according to current usage was chosen: “heat flux velocity” with the symbol u, in Russian: cкopocть пoтoкa тeплa]. This unusual use of \(u_{*}\) may be confusing to the reader. Note that this also differs from Lettau’s definition of the mixing velocity \(w^{*}\).
Obukhov did not make a distinction between isotropic and non-isotropic turbulence, and so for the Richardson number it follows that
where 1/a is the turbulent Prandtl number according to our present-day definition. The derivative with respect to height is
The value of the derivative at z = 0 was defined by Obukhov as the scale of the height of the surface layer with dynamic turbulence
with a = 1. For a discussion of the use of the turbulent Prandtl number and the von Kármán constant by different authors, see Foken (2006). The Obukhov length is significantly greater than the height of the surface layer for neutral stratification. A physical interpretation of L was made by Bernhardt (1995), i.e., “the absolute value of the Obukhov length is equal to the height of an air column in which the production (L < 0) or the loss (L > 0) of turbulence kinetic energy (TKE) by buoyancy forces is equal to the dynamic production of TKE per volume unit at any height z multiplied by z”.
Properties of the Obukhov Stability Parameter
In the early 1950s, Lettau was obviously quite convinced of his scaling velocity \(w_{a}^{*}\). However, when results of the O'Neill experiment of 1953 were published in 1957, references to \(w_{a}^{*}\) were already missing (Lettau 1957, 1990) and it was not adopted by the community (Sutton 1953). Lettau's ideas are partly contradictory to the scaling of Batchelor (1950). This was explained by Inoue (1952) in a commentary to which Lettau (1952a) replied and specified his ideas. In particular, they referred to the extension of the scaling into the laminar domain and the application of Taylor's (1935) definition of isotropic turbulence for the atmosphere. Lettau again presented his ideas in a broader context in Lettau (1952b).
It is appropriate to look again at Obukhov (1946) and Lettau (1949) with regard to the further explanations. Both authors show an almost identical picture with the dependence between Ri and z/L or x and y, respectively (Fig. 2). It is noticeable that the range of validity for z/L and y was chosen to be relatively broad (as Lettau showed in another figure, even for − 5 < y < 15 as well). In contrast, the range of validity of Monin and Obukhov's (1954) similarity theory for the surface layer today is given as − 1 < z/L < 1 when defining the Obukhov length with the von Kármán constant and without the turbulent Prandtl number (Foken 2006), whereby influences of the mixed layer and the Coriolis force are certainly present (Johansson et al. 2001; Högström et al. 2002).
It must therefore be assumed that, based on his earlier works, Lettau always had the entire boundary layer in view. It is thus not unreasonable to define a thermal mixing velocity. This idea was also used by Zilitinkevich (1971) and Betchov and Yaglom (1971) for the convective atmospheric boundary layer, summarized by Kader and Yaglom (1990), and can be seen as an anticipation of Deardorff's (1970) scaling velocity for convective conditions, even if it is defined differently as.
$$ w_{*} = \left( {\frac{{g z_{i} }}{{\theta_{v} }} \overline{{w^{\prime } \theta_{v}^{\prime } }} } \right)^{1/3} , $$
(27)
where \(z_{i}\) is the mixed-layer height, and \(\theta_{v}\) is the virtual potential temperature. Interesting is the choice of the symbol for the Deardorff velocity, which is still valid today, even if there is no explicit reference to Lettau in Deardorff’s work.
An essential application of the parameter y (or z/L) is as a stability-dependent term in the profile equations for the momentum or heat exchange. Obukhov (1946) as well as Lettau (1949) tried this by modifying the exchange coefficient. In the case of Lettau, the following equation of exchange coefficient—using Eq. 10—was formulated as
because
Physically, only the root of the smaller absolute value is of interest.
Thus, Lettau has explicitly introduced the parameter y (or z/L) into the profile equations, similarly to Rossby and Montgomery (1935) and Sverdrup (1936). However, the formulation led to considerable problems in integration, and he developed the root expression as a power series.
In contrast, Obukhov modified the exchange coefficient with a function depending on the ratio of Richardson number and critical Richardson number \({Ri}_{{{\text{cr}}}}\).
In an intermediate step, Lettau formulated a parameter \({\raise0.7ex\hbox{${\beta z}$} \!\mathord{\left/ {\vphantom {{\beta z} L}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$L$}}\). However, this parameter was first introduced into the profile equations by Monin and Obukhov (1954) as a power series \(1 - {\raise0.7ex\hbox{${\beta z}$} \!\mathord{\left/ {\vphantom {{\beta z} L}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$L$}}\), the so-called universal function of the similarity theory of Monin and Obukhov.
Lettau (1956) used his formulation (Lettau 1952b) in comparison with observations and found a clearly better agreement than with the theory of Businger (1955), who preferred a purely modified mixing-length approach with references to Rossby and Montgomery (1935) and Lettau (1949). Therefore, Businger’s approach has similarities to that of Lettau (1949) and contrasts to the statistical turbulence approaches of Batchelor (1950) and Inoue (1952).